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\begin{document}

\title{Complex Analysis}
\subtitle{Chapter 4. Complex Integration}
%\institute{SLUC}
\author{LVA}
%\date
%\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
%\date{ {2023年9月21日} }

\maketitle

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\begin{frame}{Contents 1-2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item {\color{red}Fundamental Theorems}
\begin{enumerate}
\item[1.1.] Line Integrals
\item[1.2.] Rectifiable Arcs
\item[1.3.] Line Integrals as Functions of Arcs
\item[1.4.] Cauchy's Theorem for a Rectangle
\item[1.5.] Cauchy's Theorem in a Disk
\end{enumerate}
 
\item {\color{red}Cauchy's Integral Formula}
\begin{enumerate}
\item[2.1.] The Index of a Point with Respect to a Closed Curve
\item[2.2.] The Integral Formula
\item[2.3.] Higher Derivatives
\end{enumerate}

\end{enumerate}

\end{frame}

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\begin{frame}{Contents 3-4}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[3.] {\color{red}Local Properties of Analytical Functions}
\begin{enumerate}
\item[3.1.] Removable Singularities. Taylor's Theorem
\item[3.2.] Zeros and Poles
\item[3.3.] The Local Mapping
\item[3.4.] The Maximum Principle
\end{enumerate}

\item[4.] {\color{red}The General Form of Cauchy's Theorem}
\begin{enumerate}
\item[4.1.] Chains and Cycles
\item[4.2.] Simple Connectivity
\item[4.3.] Homology
\item[4.4.] The General Statement of Cauchy's Theorem
\item[4.5.] Proof of Cauchy's Theorem
\item[4.6.] Locally Exact Differentials
\item[4.7.] Multiply Connected Regions
\end{enumerate}

\end{enumerate}

\end{frame}

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\begin{frame}{Contents 5-6}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}

\item[5.] {\color{red}The Calculus of Residues}
\begin{enumerate}
\item[5.1.] The Residue Theorem
\item[5.2.] The Argument Principle
\item[5.3.] Evaluation of Definite Integrals
\end{enumerate}

\item[6.] {\color{red}Harmonic Functions}
\begin{enumerate}
\item[6.1.] Definition and Basic Properties
\item[6.2.] The Mean-value Property
\item[6.3.] Poisson's Formula
\item[6.4.] Schwarz's Theorem
\item[6.5.] The Reflection Principle
\end{enumerate}

\end{enumerate}

\end{frame}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.1. Removable Singnlarities. Taylor's Theorem. \hfill 作业4C-1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red} 
Theorem 7. Suppose that $f(z)$ is analytic in the region $\Omega'$ obtained by omitting a point $a$ from a region $\Omega$. A necessary and sufficient condition that there exist an analytic function in $\Omega$ which coincides with $f(z)$ in $\Omega'$ is that 
$$
\lim\limits_{z\to a} (z-a)f(z) = 0.
$$
The extended function is uniquely determined.
}

\item  Proof. 
\begin{enumerate}
\item  Necessity. 
 
\item  Sufficiency. 

\end{enumerate}

\end{itemize}

\end{frame}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.1. Removable Singnlarities. Taylor's Theorem. \hfill 作业4C-2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red} 
Theorem 8. If $f(z)$ is analytic in a region $\Omega$, containing $a$, it is possible to write
%(28)
$$
f(z) = f(a) + \frac{f'(a)}{1!}(z-a) + \frac{f''(a)}{2!}(z-a)^2 + \cdots + 
\frac{f^{(n-1)}(a)}{(n-1)!}(z-a)^{n-1} + f_n(z)(z-a)^n,
$$
where $f_n(z)$ is analytic in $\Omega$.
}

\item  Proof. 
\begin{enumerate} 
\item 
 
\item 
 
\item 

\end{enumerate}

\end{itemize}

\end{frame}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.2. Zeros and Poles.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red} 
Theorem 9. An analytic function comes arbitrarily close to any complex value in every neighborhood of an essential singularity.
}

\item  Proof. 
\begin{enumerate}
\item 
 
\item 
 
\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.2. Zeros and Poles. Exercise - 1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
If $f(z)$ and $g(z)$ have the algebraic orders $h$ and $k$ at $z = a$, show that $fg$ has the order $h+k$, $f/g$ the order $h-k$, and $f+g$ an order which does not exceed $\max(h,k)$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.2. Zeros and Poles. Exercise - 2  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Show that a function which is analytic in the whole plane and has a nonessential singularity at $\infty$ reduces to a polynomial. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.2. Zeros and Poles. Exercise - 3  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Show that the functions $e^z$, $\sin z$ and $\cos z$ have essential singularities
at $\infty$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.2. Zeros and Poles. Exercise - 4  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Show that any function which is meromorphic in the extended plane is rational.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.2. Zeros and Poles. Exercise - 5  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Prove that an isolated singularity of $f(z)$ is removable as soon as either $\mathrm{Re}\, f(z)$ or $\mathrm{Im}\, f(z)$ is bounded above or below. 
Hint: Apply a fractional linear transformation.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.2. Zeros and Poles. Exercise - 6  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Show that an isolated singularity of $f(z)$ cannot be a pole of $\exp f(z)$.
Hint: $f$ and $e^f$ cannot have a common pole (why?). 
Now apply Theorem 9.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.3. The Local Mapping. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red} 
Theorem 10. Let $z_j$ be the zeros of a function $f(z)$ which is analytic in a disk $\Delta$ and does not vanish identically, each zero being counted as many times as its order indicates. For every closed curve $\gamma$ in $\Delta$ which does not pass through a zero
%(32)
$$
\sum\limits_{j} n(\gamma,z_j) = \frac{1}{2\pi i} \int_\gamma \frac{f'(z)}{f(z)}dz,
$$
where the sum has only a finite number of terms $\neq 0$.
}

\item  Proof. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.3. The Local Mapping. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red} 
Theorem 11. Suppose that $f(z)$ is analytic at $z_0$, $f(z_0) = w_0$, and that $f(z)-w_0$ has a zero of order $n$ at $z_0$. If $\varepsilon > 0$ is sufficiently small, there exists a corresponding $\delta > 0$ such that for all $a$ with $a-w_0|<\delta$ the equation $f(z) = a$ has exactly $n$ roots in the disk $|z-z_0|< \varepsilon$.

}

\item  Proof. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.3. The Local Mapping. Exercise - 1 \hfill 作业4C-3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Determine explicitly the largest disk about the origin whose image under the mapping $w = z^2 + z$ is one to one.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.3. The Local Mapping. Exercise - 2  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Same problem for $w = e^z$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.3. The Local Mapping. Exercise - 3  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Apply the representation $f(z) = w_0 + \zeta(z)^n$ to $\cos z$ with $z_0 = 0$.
Determine $\zeta(z)$ explicitly.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.3. The Local Mapping. Exercise - 4  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
If $f(z)$ is analytic at the origin and $f'(O) \neq 0$, prove the existence of
an analytic $g(z)$ such that $f(z^n) = f(0) + g(z)^n$ in a neighborhood of 0.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.4. The Maximum Principle. Theorem 12.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
If $f(z)$ is analytic and nonconstant in a region $\Omega$, then its absolute value $|f(z)|$ has no maximum in $\Omega$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.4. The Maximum Principle. Theorem 13. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
If $f(z)$ is analytic for $|z| < 1$ and satisfies the conditions 
$|f(z)|\le 1$, $f(0)=0$, then $|f(z)|\le |z|$ and $|f'(0)|\le 1$. 
If $|f(z)| = |z|$ for some $z\neq 0$, or if $|f'(0)|=1$, then $f(z) = cz$ with a  constant $c$ of absolute value $1$. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.4. The Maximum Principle. Exercise - 1  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Show by use of (36), or directly, that $|f(z)|\le 1$ for $|z| \le 1$ implies
$$\frac{f'(z)}{(1-|f(z)|^2)} \le \frac{1}{1-|z|^2}. $$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.4. The Maximum Principle. Exercise - 2  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
If $f(z)$ is analytic and $\mathrm{Im}\, f(z) \ge 0$ for $\mathrm{Im}\, z > 0$, show that
$$ \frac{f(z)-f(z_0)}{f(z)-\overline{f(z_0)}} \le \frac{z-z_0}{z-\overline{z_0}}$$
and $$\frac{|f'(z)|}{\mathrm{Im}f(z)} \le \frac{1}{y}, \,\, (z=x+iy). $$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.4. The Maximum Principle. Exercise - 3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
In Ex. 1 and 2, prove that equality implies that $f(z)$ is a linear transformation.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.4. The Maximum Principle. Exercise - 4 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Derive corresponding inequalities if $f(z)$ maps $|z| < 1$ into the upper half plane.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{3.4. The Maximum Principle. Exercise - 5 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Prove by use of Schwarz's lemma that every one-to-one conformal mapping of a disk onto another (or a half plane) is given by a linear transformation.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{4.1. Chains tind Cycles.   }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
The integral of an exact differential over any cycle is zero.
}

\item  Answer. 
\begin{enumerate}
\item A chain is a cycle if it can be represented as a sum of closed curves. 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{4.2. Simple Connectivity.   }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
What is a simply connected region?
}

\item  Answer. 
\begin{enumerate}
\item A region is simply connected if its complement with respect to the extended plane is connected.
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{4.2. Simple Connectivity. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red} 
Theorem 14. A region $\Omega$ is simply connected if and only if $n(\gamma,a) = 0$ for all cycles $\gamma$ in $\Omega$ and all points $a$ which do not belong to $\Omega$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{4.3. Homology. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
When is a cycle said to be homologous to zero?
}

\item  Answer. 
\begin{enumerate}
\item 
A cycle $\gamma$ in an open set $\Omega$ is said to be homologous to zero with respect to $\Omega$ if $n(\gamma,a) = 0$ for all points $a$ in the complement of $\Omega$.

\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{4.4. The General Statement of Cauchy's Theorem. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red} 
Theorem 15. If $f(z)$ is analytic in $\Omega$, then
$$
\int_\gamma f(z) dz = 0
$$
for every cycle $\gamma$ which is homologous to zero in $\Omega$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{4.5. Proof of Cauchy's Theorem.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.

}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{4.6. Locally Exact Differentials. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red} 
Theorem 16. If $p dx + q dy$ is locally exact in $\Omega$, then 
$$\int_\gamma p dx + q dy = 0$$
for every cycle $\gamma\sim 0$ in $\Omega$.
}

\item  Proof. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{4.7. Multiply Connected Regions.   }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
What is finite connectivity and infinite connectivity? 
}

\item  Answer. 
\begin{enumerate}
\item A region which is not simply connected is called multiply connected. More precisely, $\Omega$ is said to have the finite connectivity $n$ if the complement of $\Omega$ has exactly $n$ components and infinite connectivity if the complement has infinitely many components. 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{4.7. Multiply Connected Regions. Exercise - 1  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Prove without use of Theorem 16 that $p dx + q dy$ is locally exact in $\Omega$ if and only if $$\int_{\partial R} p dx + qdy = 0$$
for every rectangle $R \subset \Omega$ with sides parallel to the axes. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{4.7. Multiply Connected Regions. Exercise - 2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Prove that the region obtained from a simply connected region by removing $m$ points has the connectivity $m+1$, and find a homology basis. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{4.7. Multiply Connected Regions. Exercise - 3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Show that the bounded regions determined by a closed curve are simply connected,  while the unbounded region is doubly connected. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{4.7. Multiply Connected Regions. Exercise - 4 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Show that single-valued analytic branches of $\log z$, $z^a$ and $z^z$ can be 
defined in any simply connected region which does not contain the origin.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{4.7. Multiply Connected Regions. Exercise - 5 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Show that a single-valued analytic branch of $\sqrt{1-z^2}$ can be defined in any region such that the points $\pm 1$ are in the same component of the complement. 
What are the possible values of
$$\int\frac{dz}{\sqrt{1-z^2}}$$
over a closed curve in the region?
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{5.1. The Residue Theorem. Definition 3. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
What is the residue of an analytic function at an isolated singularity?
}

\item  Answer. 
\begin{enumerate}
\item 
The residue of $f(z)$ at an isolated singularity $a$ is the unique complex number $R$ which makes $f(z) - R/(z-a)$ the derivative of a single-valued analytic function in an annulus $0 < |z -a| < \delta$.

\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{5.1. The Residue Theorem. \hfill 作业4E-1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red} 
Theorem 17. Let $f(z)$ be analytic except for isolated singularities $a_j$ in a region $\Omega$.
Then
%(47)
$$
\frac{1}{2\pi i} \int_\gamma f(z)dz = \sum\limits_{j} n(\gamma,a_j)\mathrm{Res}_{z=a_j}f(z) 
$$
for any cycle $\gamma$ which is homologous to zero in $\Omega$ and does not pass through any of the points $a_j$.
}

\item  Proof. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.1. The Residue Theorem. Definition 4. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.

}

\item  Answer. 
\begin{enumerate}
\item 
A cycle $\gamma$ is said to bound the region $\Omega$ if and only if $n(\gamma,a)$ is defined and equal to 1 for all points $a \in \Omega$ and either undefined or equal to zero for all points $a$ not in $\Omega$.

\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.2. The Argument Principle. \hfill 作业4E-2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red} 
Theorem 18. If $f(z)$ is meromorphic in $\Omega$ with the zeros $a_j$ and the poles $b_k$, then
%(48)
$$
\frac{1}{2\pi i} \int_\gamma \frac{f'(z)}{f(z)}dz = \sum\limits_{j} n(\gamma,a_j)
- \sum\limits_{k} n(\gamma,b_k)
$$
for every cycle $\gamma$ which is homologous to zero in $\Omega$ and does not pass through any of the zeros or poles.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.2. The Argument Principle. \hfill 作业4E-3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red} 
Rouche's theorem. Let $\gamma$ be homologous to zero in $\Omega$ and such that $n(\gamma,z)$ is either 0 or 1 for any point $z$ not on $\gamma$. 
Suppose that $f(z)$ and $g(z)$ are analytic in $\Omega$ and satisfy the inequality $|f(z)-g(z)| < |f(z)|$ on $\gamma$. Then $f(z)$ and $g(z)$ have the same number of zeros enclosed by $\gamma$. 
}

\item  Proof. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.2. The Argument Principle. Exercise - 1 \hfill 作业4E-4 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
How many roots does the equation 
$z^7 - 2z^5 + 6z^3 - z + 1 = 0$ 
have in the disk $|z| < 1$? Hint: Look for the biggest term when $|z|=1$ 
and apply Rouche's theorem. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.2. The Argument Principle. Exercise - 2  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
How many roots of the equation $z^4 - 6z + 3 = 0$ have their modulus between 1 and 2?
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.2. The Argument Principle. Exercise - 3 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
How many roots of the equation $z^4 + 8z^3 + 3z^2 + 8z + 3 = 0$ lie in the right half plane? Hint: Sketch the image of the imaginary axis and apply the argument principle to a large half disk.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.3. Evaluation of Definite Integrals. Example 1. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
All integrals of the form
$$\int_0^{2\pi} R(\cos\theta, \sin\theta) d\theta $$
where the integrand is a rational function of $\cos\theta$ and $\sin\theta$ can be easily evaluated by means of residues.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.3. Evaluation of Definite Integrals. Example 2. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
An integral of the form
$$\int_{-\infty}^{\infty} R(x)dx$$
converges if and only if in the rational function $R(x)$ the degree of the denominator is at least two units higher than the degree of the numerator, and if no pole lies on the real axis. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.3. Evaluation of Definite Integrals. Example 3.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
The same method can be applied to an integral of the form
$$\int_{-\infty}^{\infty} R(x)e^{ix} dx.  $$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.3. Evaluation of Definite Integrals. Example 4.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
The next category of integrals have the form
$$ \int_0^{\infty} x^\alpha R(x) dx $$
where the exponent $\alpha$ is real and may be supposed to lie in the interval $0 < \alpha < 1$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.3. Evaluation of Definite Integrals. Example 5. \hfill 作业4E-5 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
As a final example we compute the special integral
$$ \int_0^{\pi} \log \sin \theta d\theta $$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.3. Evaluation of Definite Integrals. Exercise 1. \hfill 作业4E-6 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Find the poles and residues of the following functions:
\begin{enumerate}
\item  $\frac{1}{z^2+5z+6}$. 
\item  $\frac{1}{(z^2-1)^2}$. 
\item  $\frac{1}{\sin z}$. 
\item  $\cot z$. 
\item  $\frac{1}{\sin^2 z}$. 
\item  $\frac{1}{z^m(1-z)^n}$ ($m, n$ positive integers).
\end{enumerate}
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.3. Evaluation of Definite Integrals. Exercise 2. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Show that in Sec. 5.3, Example 3, the integral may be extended over a right-angled isosceles triangle. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.3. Evaluation of Definite Integrals. Exercise 3. \hfill 作业4E-7 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Evaluate the following integrals by the method of residues:
\begin{enumerate}
\item  $\int_0^{\pi/2}\frac{dx}{a+\sin^2 x}, |a|>1$.  
\item  $\int_0^{\infty}\frac{x^2dx}{x^4+5x^2+6}$.  
\item  $\int_0^{\infty}\frac{x^2-x+2}{x^4+10x^2+9}dx$.  
\item  $\int_0^{\infty}\frac{x^2dx}{(x^2+a^2)^3}$, $a$ real.  
\item  $\int_0^{\infty}\frac{\cos x}{x^2+a^2}dx$, $a$ real.  
\item  $\int_0^{\infty}\frac{x\sin x}{x^2+a^2}$, $a$ real.  
\item  $\int_0^{\infty}\frac{x^{1/3}}{1+x^2}$.  
\item  $\int_0^{\infty}\frac{\log x}{1+x^2}dx$.  
\item  $\int_0^{\infty}\frac{\log (1+x^2)}{x^{1+\alpha}}dx, (0<\alpha<2)$.  
(Try integration by parts.)
\end{enumerate}
}

%\item  Answer. 
%\begin{enumerate}
%\item 
% 
%\item 
%
%\item 
%
%\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.3. Evaluation of Definite Integrals. Exercise 4. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question.
Compute
$$\int_{|z|=\rho} \frac{|dz|}{|z-a|^2}, |a|\neq \rho. $$
Hint: Use $z\bar{z} = \rho^2$ to convert the integral to a line integral of a rational function.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{5.3. Evaluation of Definite Integrals. Exercise 5. \hfill 作业4E-8 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}(Bergman's kernel formula) 
Complex integration can sometimes be used to evaluate area integrals. 
As an illustration, show that if $f(z)$ is analytic and bounded for $|z| < 1$ and if $|\zeta| < 1$, then 
$$f(\zeta) = \frac{1}{\pi} \iint_{|z|<1} \frac{f(z)dxdy}{(1-\bar{z}\zeta)^2}. $$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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